The original approach for the self-consistent inclusion of gravity into quantum mechanics of a particle is developed. (There are no connections with second quantization.) The nonstandard action principle is constructed for the stationary situation: the quantum particle in a stationary state creating some nonclassical stationary gravitational field and interacting with it, The accompanying problem of covariantization of quantum operators is considered. The general theory is illustrated by the Newtonian-Schrödingerian and quasi classical limiting cases. The levels of applicability of ordinary quantum mechanics and the problems of measurements and interpretation of nonclassical gravity are discussed. The “uncertainty relations” connecting uncertainties of some “local” parts of curvature and those of the particle’s position and momentum are derived. The superposition principle is generalized on the base of some approximate action.
The original approach for the self-consistent inclusion of gravity into quantum mechanics of a particle is developed. (There are no connections with second quantization.) The nonstandard action principle is constructed for the stationary situation: the quantum particle in a stationary state creating some nonclassical stationary gravitational field and interacting with it, The accompanying problem of covariantization of quantum operators is considered. The general theory is illustrated by the Newtonian-Schrödingerian and quasi classical limiting cases. The levels of applicability of ordinary quantum mechanics and the problems of measurements and interpretation of nonclassical gravity are discussed. The “uncertainty relations” connecting uncertainties of some “local” parts of curvature and those of the particle’s position and momentum are derived. The superposition principle is generalized on the base of some approximate action.
We develop a nonlinear quantum theory of Newtonian gravity consistent with an objective interpretation of the wavefunction. Inspired by the ideas of Schrodinger, and Bell, we seek a dimensional reduction procedure to map complex wavefunctions in configuration space onto a family of observable fields in space-time. Consideration of quasi-classical conservation laws selects the reduced one-body quantities as the basis for an explicit quasi-classical coarse-graining. These we interpret as describing the objective reality of the laboratory. Thereafter, we examine what may stand in the role of the usual Copenhagen observer to localise this quantity against macroscopic dispersion. Only a tiny change is needed, via a generically attractive self-potential. A nonlinear treatment of gravitational self-energy is thus advanced. This term sets a scale for all wavepackets. The Newtonian cosmology is thus closed, without need of an external observer. Finally, the concept of quantisation is re-interpreted as a nonlinear eigenvalue problem. To illustrate, we exhibit an elementary family of gravitationally self-bound solitary waves. Contrasting this theory with its canonically quantised analogue, we find that the given interpretation is empirically distinguishable, in principle. This result encourages deeper study of nonlinear field theories as a testable alternative to canonically quantised gravity.
The stability of a quantum superposition f two different stationary mass distributions is examined, where the perturbing effect of each distribution on the space-time structure is taken into account, in accordance with the principles of general relativity. It is argued that the definition of the time-translation operator for the superposed space-times involves an inherent ill-definedness, leading to an essential uncertainty in the energy of the superposed state which, in the Newtonian limit, is proportional to the gravitational self-energy EA of the difference between the two mass distributions. This is consistent with a suggested finite lifetime of the order of?t/ELx for the superposed state, in agreement with a certain proposal made by the author for a gravitationally induced spontaneous quantum state reduction, and with closely related earlier suggestions by Di6si and by Ghirardi et al. 1. THE PROBLEM OF QUANTUM STATE REDUCTION The fundamental problem of quantum mechanics, as that theory is presently understood, is to make sense of the reduction of the state vector (i.e. collapse of the wavefunction), denoted here by R. This issue is usually addressed in terms of the "quantum measurement problem", which is to comprehend how, upon measurement of a quantum system, this (seem-ingly) discontinuous R-process can come about. A measurement, after all, merely consists of the quantum state under consideration becoming en-tangled with a more extended part of the physical universe, e.g. with a
The possibility that a classical space-time and quantum matter cohabit at the deepest level, i.e., the possibility of having a fundamental and not phenomenological semiclassical gravity, is often disregarded for lack of a good candidate theory. The standard semiclassical theory suffers from fundamental inconsistencies (e.g., Schr\"odinger cat sources, faster-than-light communication and violation of the Born rule) which can only be ignored in simple typical situations. We harness the power of spontaneous localization models, historically constructed to solve the measurement problem in quantum mechanics, to build a consistent theory of (stochastic) semiclassical gravity in the Newtonian limit. Our model makes quantitative and potentially testable predictions: we recover the Newtonian pair potential up to a short distance cutoff (hence, we predict no one-particle self-interaction) and uncover an additional gravitational decoherence term which depends on the specifics of the underlying spontaneous localization model considered. We hint at a possible program to go past the Newtonian limit, towards a consistent general relativistic semiclassical gravity.
A proper deformation of the underlying coordinate and momentum commutation relations in quantum mechanics provides a phenomenological approach to account for the influence of gravity on small scales. Introducing the squared momentum term results in a generalized uncertainty principle, which limits the minimum uncertainty in particle position to the Planck length. However, such a deformation of the commutator significantly changes the formalism, making it separate from the canonical formalism of quantum mechanics. In this study, it is shown that the deformed algebra of position and momentum operators can be incorporated into the framework of ordinary quantum mechanics.
Abstract Inspired by the generalized uncertainty principle, which adds gravitational effects to the standard description of quantum uncertainty, we extend the exact uncertainty principle approach by Hall and Reginatto (2002 J. Phys. A: Math. Gen. 35 3289 ), and obtain a (quasi)nonlinear Schrödinger equation. This quantum evolution equation of unusual form, enjoys several desired properties like separation of non-interacting subsystems or plane-wave solutions for free particles. Starting with the harmonic oscillator example, we show that every solution of this equation respects the gravitationally induced minimal position uncertainty proportional to the Planck length. Quite surprisingly, our result successfully merges the core of classical physics with non-relativistic quantum mechanics in its extremal form. We predict that the commonly accepted phenomenon, namely a modification of a free-particle dispersion relation due to quantum gravity might not occur in reality.
Quantum mechanics at present cannot be unified with the theory of gravity at the deepest level, and to guide research towards the solution of this fundamental problem, we need to look for ways to observe or refute predictions originating from attempts to combine quantum theory with gravity. The influence of the gravitational field created by the material density given by the wave function itself gives rise to nontrivial phenomena. In this study I consider the wave function for the center-of-mass coordinate of a spherical mass distribution under the influence of the self-interaction of Newtonian gravity. I solve numerically for the ground state in the presence of an unstable potential and find that the energy of the free-space bound state can be lowered despite the nontrapping character of the potential. The center-of-mass ground state becomes increasingly localized for the used unstable potentials, although only in a limited parameter regime. The feebleness of the energy shift makes the observation of these effects demanding and requires further developments in the cooling of material particles. In addition, the influence of gravitational perturbations that are present in typical laboratory settings necessitates the use of extremely quiet and controlled environments such as those provided by recently proposed space-borne experiments.